Transmission characteristics of planetary gear wear in multistage gear transmission system

2.1. Fault simulation test rig of a multistage gear transmission system

The multistage gear transmission system fault simulation test rig is taken as the research object, shown in Fig. 1. The vibration characteristic analysis and the transmission characteristic analysis are carried out.

As can be seen from Fig. 1, there are two gearboxes in the gear transmission system test rig. They are the fixed-axis gearbox made up of two-stage fixed-axis gears and the planetary gearbox made up of one-stage planetary gear train. The system structure diagram is shown in Fig. 2, where, $T_{in}$ is the input torque, driven by three-phase AC asynchronous motor; $T_{out}$ is the output torque, which can be set through the electromagnetic powder brake; gears 1 and 2 form the high-speed gear pair; gear 1 is the high-speed driving gear; gear 2 is the high-speed driven gear; gears 3 and 4 are the medium-speed driving gear and driven gear respectively; the planetary gear train is the low-speed stage, $p$ is the planetary gear, $s$ is the sun gear, $c$ is the planet carrier, and $r$ is the ring gear (fixed). The gears in the test rig are all involute cylindrical spur gears with pressure angle $\alpha$ of 20°. Gear parameters are shown in Table 1.

Fig. 1Fault simulation test rig of a multistage gear transmission system: 1 – motor, 2 – torque sensor and encoder, 3 – two stage fixed-axis gearbox, 4 – radial load of bearing, 5 – one stage planetary gearbox, 6 – brake

Fig. 2Structural diagram of the gear transmission system test rig

2.2. Dynamic model of gearboxes

Taking the fault simulation test rig of the multistage gear transmission system as the prototype, the corresponding UG 3D solid model was established. The gear, axis, box and other components were built, and gearboxes were integrally assembled by applying constraint relations. The assembled body and its internal structure are shown in Fig. 3.

The UG model was imported into ADAMS for dynamic simulation. According to the motion mode, the corresponding motion constraints, drive and load were added to the gear box. Contact forces were applied between meshing gears to simulate their meshing relations. Contact forces were set by stiffness, force exponent, damping, and penetration depth.

Fig. 3Multistage gear transmission system test rig solid model: a) assembled body, and b) internal structure

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Contact stiffness $K$ is affected by the material properties and the contact surface shape. The calculation formula is as follows:

where, $R_1$, $R_2$ are the curvature radiuses of meshing tooth profile. Since the curvature radius of involute gear meshing point changes in the meshing process. The calculation point is taken as: $R_i=m{z}_i\mathrm{s}\mathrm{i}\mathrm{n}\alpha /2$, $i=$ 1, 2; $h_1$ and $h_2$ are material parameters, $h_i=\left(1-u_i^2\right)/\pi E_i$, $i=$ 1, 2; $u$ and $E$ are the Poisson’s ratio and elastic modulus of the material, respectively.

According to the equivalent energy loss, the damping formula is as follows:

where, $e$ is the elastic recovery coefficient, whose value is usually determined by experiments; $\delta$ is the penetration depth; $n$ is the nonlinear exponential; $U$ is the collision velocity.

Force exponent and penetration depth can be selected from the system default values first.

Considering nonlinear characteristics of the gearboxes, the fixed axis gearbox and the planetary gearbox were softened [21, 22] to obtain more real gear dynamics characteristics.

The contact forces of the meshing parts of fixed-axis gears and planetary gears were calculated by ADAMS. The simulation results were compared with the results of dynamic differential equations and experimental data, so as to modify some of the set parameters after softening. The final settings are shown in Fig. 4.

3.1. Analysis of gear contact force signals under normal state

The dynamic simulation of the system under normal state was carried out. The contact force simulation signals of high-speed, medium-speed and low-speed external meshing (planetary gear meshing with sun gear) and low-speed internal meshing (planetary gear meshing with ring gear) were extracted respectively, as shown in Fig. 5.

As can be seen from Figs. 5(a) and (b), the waveforms of high-speed and medium-speed signals are relatively dense, mainly composed of high frequency component. In Figs. 5(c) and (d), the amplitude of low-speed contact force signal is large. Since the positions of meshing points of low-speed external meshing and low-speed internal meshing change with the rotation of the planet carrier, vibration transfer paths between meshing points and boxes are changed. The passing effect of planetary gear will cause the amplitude modulation of meshing vibration. The planet carrier rotation frequency (1.015 Hz) is the modulation frequency. The small fluctuation in Fig. 5(e) is the low-speed meshing frequency (101.5 Hz).

In order to observe the frequency components and the proportion of signals, the power spectrum corresponding to each time domain signal was obtained, as shown in Fig. 6.

In high-speed signal in Fig. 6(a), the vibration frequency is dominated by high-speed meshing frequency $f_1$ (1160 Hz). Medium-speed meshing frequency $f_2$ (417.6 Hz) and low-speed meshing frequency $f_3$ (101.5 Hz) are relatively weak and hardly visible.

In medium-speed signal in Fig. 6(b), in addition to medium-speed meshing frequency $f_2$ (417.6 Hz), there are also high-speed meshing frequency $f_1$ (1160 Hz), low-speed meshing frequency $f_3$ (101.5 Hz) and its twice $\text{2}{f}_3$ (203 Hz). It shows the correlation between various gear signals.

Fig. 5Time domains of gear contact force simulation signal under normal state: a) high-speed, b) medium-speed, c) low-speed external meshing, d) low-speed internal meshing, e) low-speed external meshing magnification

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Fig. 6Frequency domains of gear contact force simulation signal under normal state: a) high-speed, b) medium-speed, c) low-speed external meshing, d) low-speed internal meshing

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In Figs. 6(c) and (d), high-speed meshing frequency $f_1$ (1160 Hz) and medium-speed meshing frequency $f_2$ (417.6 Hz) attenuate obviously. The planet carrier rotation frequency $f_c$(1.015 Hz) has the highest amplitude, followed by the frequency components are mainly low-speed meshing frequency $f_3$ (101.5 Hz) and its higher order harmonics ($\text{2}{f}_3$, $\text{3}{f}_3$, ...). The response frequency amplitude of low-speed external meshing in Fig. 6(c) is larger than that of low-speed internal meshing in Fig. 6(d).

The above time domain and frequency domain charts conform to the basic law of the gear vibration and verify the correctness of the model.

3.2. Analysis of gear contact force signals under planetary gear wear state

The gear thickness of the test rig was measured by the gear thickness measuring instrument. The thickness at the gear node was measured to be 1.3 mm, which was 0.2 mm less than before, that is, the wear loss was 0.2 mm. When gear wear exceeds 20 %, it needs to be replaced. By comparing the wear degree standard in Table 3, it can be seen that the degree of wear gear on the test rig is medium wear.

In the simulation model, one planetary gear of the low-speed stage was set to wear. The degree of wear was consistent with the wear gear in the test rig (wear loss 0.1 mm on both sides), as shown in Fig. 7. Fig. 7(a) shows the actual gear with wear, and Fig. 7(b) shows the corresponding simulation gear model with wear.

It is found that the planetary gear wear has an important influence on the low-speed internal and external meshing signals. Since the fault characteristics of the two are similar, only the low-speed external meshing signal was listed for illustration, as shown in Fig. 8.

Fig. 7Wear gear: a) actual gear with wear, b) simulation gear model with wear

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Fig. 8Gear contact force simulation signal under the state of planetary gear wear: a) low-speed external meshing time domain, b) low-speed external meshing frequency domain

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As shown in Fig. 8(a), the vibration range of low-speed signal in time domain decreases from ±1400 N to ±400 N. The fluctuation of low-speed meshing frequency $f_3$(101.5 hz) increases. In the frequency domain, comparing Fig. 6(c) with Fig. 8(b), the low-speed meshing frequency $f_3$ (101.5 Hz) increases from 4.3×10⁵ N to 1.8×10⁷ N, and its harmonic components $n{f}_3$ ($n=$ 1, 2, ...) also increases.

4.1. Test signal characteristics under normal state

The acceleration response signals on the surface of gearboxes under normal state were measured, in which the input axis frequency was 40 Hz, the output axis was loaded with 10 V, and the sensitivity of acceleration sensor was 100.2 mv/g. Considering the difference of collected signals on different boxes, the acceleration vibration response signals of fixed-axis gearbox and planetary gearbox in horizontal direction were measured separately. The test results of two boxes on the surface under normal state are shown in Fig. 9.

It can be seen from Fig. 9 that the main frequency components of two boxes in normal state are high-speed meshing frequency $f_1$ (1160 Hz), high-speed axis frequency $f_d$ (40 Hz), medium-speed meshing frequency $f_2$ (417.6 Hz) and its twice $\text{2}{f}_2$ (835 Hz), low-speed meshing frequency $f_3$ (101.5 Hz), etc.

Affected by distance, the signal intensities of two boxes are different. In Fig. 9(a), high-speed axis frequency $f_d$ (40 Hz) and high-speed meshing frequency $f_1$ (1160 Hz) are significantly weakened because of far away from high-speed stage. The amplitude of low-speed meshing frequency $f_3$ (101.5 Hz) is enhanced.

Fig. 9Power spectrum signals of the test rig under normal state: a) planetary gearbox, b) fixed axis gearbox

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4.2. Transfer function of each transfer path under normal state

The vibration transfer path system model of multistage gear transmission system is established by system identification method. Take the four contact forces in Fig. 6 as the input excitation forces, and the test signals on the surface of gearboxes in Fig. 9 as the output. The auto-regressive exogenous (ARX) model with multi-point input and single point output and Tustin inverse transformation were used to solve the problem. The amplitude-frequency response curves of transfer functions of meshing excitation forces transmitted to fixed-axis gearbox and planetary gearbox under normal state are shown in Fig. 10.

In Fig. 10, 40 Hz has the highest amplitude, and the frequency is high-speed axis frequency. Therefore, high-speed stage contact force has the greatest contribution to the frequency. Medium-speed and low-speed stage also has some contributions. Meshing frequencies of $f_1$, $f_2$ and $f_3$ are not obvious. But $\text{2}{f}_2$ (835 Hz) is obvious, which is the contribution of low-speed internal and external meshing process [23].

Fig. 10The amplitude-frequency response curve of transfer function of each transfer path under normal state: a) contact force is transmitted to planetary gearbox, b) contact force is transmitted to fixed axis gearbox

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In Fig. 10(a), planetary gearbox is far away from high-speed stage, signal intensity of high-speed stage is weakened, and low-speed signal is enhanced. The amplitude of high-speed axis rotation frequency $f_d$ (40 Hz) decreases. The frequency characteristic of low-speed meshing frequency $f_3$ (101.5 Hz) increases. Low-speed internal and external meshing also causes a large value of $\text{2}{f}_2$ (835 Hz).

4.3. Test signal characteristics of planetary gear wear

A normal planetary gear was replaced with a worn planetary gear in Fig. 7(a). Test signals of two boxes are shown in Fig. 11.

Fig. 11Power spectrum signals of the test rig under planetary gear wear: a) planetary gearbox, b) fixed axis gearbox

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As shown in Fig. 11(a), under the state of planetary gear wear, the amplitude of 400 Hz is increased remarkably, which is much higher than other frequency components in the spectrum. Due to the weak signal of the planet gears, the peak of the planet gear meshing frequency in the low frequency part is not obvious. In Fig. 11(b), the amplitude of 400 Hz under the state of planetary gear wear is also affected by the connection of the planetary gearbox, which is much higher than other frequency components. Because the signal of the fixed axis gearbox is strong, the other frequency components are clear. It can be seen that the wear fault of planetary gear caused the amplitude of 400 Hz to rise, and transmitted the characteristic to the fixed axis gear.

4.4. Transfer function of each transfer path under the state of planetary gear wear

Taking the contact force signals of high-speed and medium-speed in Fig. 6 and low speed under planetary gear wear state in Fig. 8(b) as the input excitation forces. Taking the test fault signals on the surface of the gearbox in Fig. 11 as the output. The amplitude-frequency response curves of transfer functions of meshing excitation forces transmitted to fixed-axis gearbox and planetary gearbox under planetary gear wear state are shown in Fig. 12.

Fig. 12The amplitude-frequency response curve of transfer function of each transfer path under planetary gear wear: a) contact force is transmitted to the planetary gearbox, b) contact force is transmitted to the fixed axis gearbox

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Overall, whether it is the planetary gearbox or the fixed-axis gearbox, the signal is enhanced around 400 Hz. At other frequencies, there is a significant weakening that causes other frequency components in the planetary gearbox to be invisible. The 400 Hz frequency component is neither the meshing frequency of the system nor the fault frequency of the planetary gear, so it is necessary to explore its source.

In the author’s study of the nonlinear dynamics of the system [24], it was found that the bifurcation characteristics of the system will change when the gear clearance changes, as showed in Fig. 13. According to Chinese national standard GB 2363-90, the calculating formulas of side clearance in engineering application can be obtained. After calculating, the available interval of dimensionless clearance ${\stackrel{-}{e}}_{ai}$ for meshing gear is [0.1, 1], corresponding to 0-20 % gear wear. When ${\stackrel{-}{e}}_{ai}$ increases gradually in the interval [0.1, 1], the single-period motion (Fig. 13(a)) change into a three-period motion (Fig. 13(e)), and then 1/3 sub-harmonic resonance occurs. The sub-harmonic resonance frequency $f_1/\text{3}$ (1160/3 Hz) appears in the test signal. It can be seen that the condition of resonance is that the dimensionless clearance ${\stackrel{-}{e}}_{ai}$ is greater than 0.5, that is, the wear loss is greater than 0.167 mm.

As can be seen from Fig. 14, the amplitude of $f_1/\text{3}$ has exceeded all the peaks when resonance occurs. The sub-harmonic resonance frequency is related to the high speed stage, so it comes from the high speed stage. In Fig. 12, the wear failure causes the gear tooth clearance to increase, and the wear loss of 0.2 mm is close to ${\stackrel{-}{e}}_{ai}=$ 0.6 in the simulation, which is in the sub-harmonic resonance range. Therefore, the amplitude of 400 Hz is the highest, and the contribution of high-speed stage is the largest. This conclusion is consistent with the author's nonlinear research results.

417.6 Hz is the meshing frequency of the medium-speed gear, which resonates with $f_1/\text{3}$. Therefore, the medium-speed stage in Fig. 12 also contributes to this frequency.

400 Hz mainly comes from the high speed sub-harmonic resonance, followed by the medium speed meshing frequency. From the amplitude, we can see that the amplitude of 400 Hz in Fig. 12(b) is larger than that in Fig. 12(a).

Fig. 13The Poincaré section of the high-speed gear meshing point: a) e¯ai= 0.1, b) e¯ai= 0.2, c) e¯ai= 0.3, d) e¯ai= 0.4, e) e¯ai= 0.5, f) e¯ai= 0.6, g) e¯ai= 0.8, h) e¯ai= 0.9, i) e¯ai= 1

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It can be seen that the sub-harmonic resonance is caused by the wear of the planetary gear, but the frequency is related to the meshing frequency of the high-speed stage, so it is produced by the high-speed stage and transmitted to the planetary gearbox.

Fig. 14High-speed spectrum at e¯ai= 0.6

In order to verify the correctness of the results obtained in this paper, the research results of other scholars are compared with those of this paper. Feng [25, 26] has the same test-bed as this paper. In the spectrum of their experimental signals, it is found that wear faults cause a sharp increase in the third harmonic amplitude, which is not found in other faults. We know that the occurrence of sub-harmonic resonance mostly comes from rotor rub, and the main characteristics of rotor rub are that the frequency includes second-harmonic, third-harmonic and 1/2-harmonic and 1/3-harmonic components. It is also proposed in paper [27] that rotor rub excites forward and backward whirling frequency components almost equally. Therefore, in Feng’s experimental signals, 1/3 sub-harmonic resonance characteristics are reflected in the third harmonic. They only mentioned that the higher order harmonics increase in amplitude, but they do not explain why the third harmonic far exceeds the others. The results of this study are similar to those of Feng’s and explain the unspecified parts of his research results from the point of view of non-linearity. It shows that the results of this study have common characteristics.